Download Chaotic mixing in a Joule-heated glass melt

PHYSICS OF FLUIDS 22, 013101 共2010兲
Chaotic mixing in a Joule-heated glass melt
Sugilal Gopalakrishnan,1,a兲 André Thess,1,b兲 Günter Weidmann,2 and Ulrich Lange2
1
Institute of Thermodynamics and Fluid Mechanics, Ilmenau University of Technology,
P.O. Box 100565, 98684 Ilmenau, Germany
2
Mathematical Simulation and Optimization, Corporate Service Melting Technology,
Schott AG, Hattenbergstrasse 10, 55122 Mainz, Germany
共Received 29 January 2009; accepted 1 December 2009; published online 4 January 2010兲
A numerical study of two-dimensional thermal convection of a highly viscous fluid driven by
volumetric heating has been carried out to investigate the effect of time-dependent thermal forcing
on the degree of mixing. The problem is relevant to electrically heated glass melting furnaces which
are traditionally operated with time-independent heating. The numerical computations carried out
are for two model fluids representing semitransparent and opaque melts, respectively. Lagrangian
motion of passive tracers is numerically simulated and degree of mixing in the glass melt is
quantified in terms of mixing entropy. The computed flow patterns indicate that time-dependent
streamlines intersect transversely and, thereby, result in well-mixed regions. The stirring mechanism
involves the rotational stretching and folding produced by two oscillating vortices with varying size
and circulation. The model predictions indicate that the chaotic mixing is strongly dependent on the
period of electrode firing cycle. The present investigation demonstrates that time-dependent thermal
forcing improves the mixing characteristics of both semitransparent and opaque glass melts
significantly. © 2010 American Institute of Physics. 关doi:10.1063/1.3280045兴
I. INTRODUCTION
Electric glass furnaces are increasingly used in the glass
industry by virtue of their advantages such as high efficiency,
small size, and ecofriendliness.1 In an all-electric glass furnace, a system of electrodes immersed in the glass melt produces thermal energy based on the principle of Joule heating.
Density gradients in the internally heated glass melt give rise
to free convection flows. In an internally heated fluid layer,
the flow can be steady or unsteady depending on its size,
physical properties, and volumetric heat source. Although
strongly turbulent flow does not occur in highly viscous molten glass, the convective transport processes in the melt have
significant influence on the quality of the glass produced.
The challenge of producing high quality glass is to avoid
streakiness which may be of thermal or chemical origin. Mechanical stirring is usually employed to avoid such inhomogeneities in the glass melt.2
Mixing in nonturbulent flows can be greatly enhanced by
chaotic advection, which is an intermediate regime between
laminar and turbulent advection.3 In two-dimensional unsteady flows and three-dimensional flows, fluid-particle trajectories can diverge exponentially and advection of individual particles in the fluid can become chaotic. In chaotic
mixing, various layers of fluid are bent and folded repeatedly
so as to bring initially distant parts close together and separate initially close parts exponentially farther. As a result, the
interfacial area between the fluid layers increases, whereas
the thickness of each layer or striation decreases. These
a兲
Also at Nuclear Recycle Group, Bhabha Atomic Research Centre, Mumbai
400085, India.
b兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected].
1070-6631/2010/22共1兲/013101/12/$30.00
complementary actions improve the mixing process substantially. The theory of chaotic mixing has been well developed
due to the pioneering work of Aref and co-workers as well as
Ottino and co-workers.4 A thorough review of the key concepts in chaotic mixing can be found in Ref. 5. Chaotic advection finds several applications in fluid mixing and material processing. In both micro- and macromixing devices,
chaotic mixing is accomplished by either geometric or temporal manipulation of flow. Electrically heated glass melting
furnaces are generally operated with time-independent heating 共steady thermal forcing兲. The goal of the present paper is
to investigate the question whether chaotic mixing induced
by time-dependent heating 共time-dependent thermal forcing兲
improves the homogenization of glass melts.
Heat transport in internally heated liquids under different
natural convection regimes ranging from steady-laminar to
turbulent flows was extensively investigated by many researchers in the past.6,7 However, literature on chaotic particle transport in fluids with internal heat generation is essentially limited to studies related to mixing in Earth’s mantle
by thermal convection.8 Reference 8 shows that the efficiency of mass transport in the Earth’s mantle is enhanced
mainly due to the increased spatiotemporal complexity of the
time-dependent flow. Motivated by the above mentioned features of chaotic advection, a numerical study was carried out
to quantify the effect of temporal manipulation of glass melt
flow on the chaotic strength and mixing in a Joule-heated
glass melt in a rectangular cavity with an aspect ratio of 2.
Results from this study are expected to provide useful information for further improvements in electric glass furnace design in order to achieve better glass quality.
The paper is organized as follows. The formulation of
the problem is presented in Sec. II and the implementation of
22, 013101-1
© 2010 American Institute of Physics
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Gopalakrishnan et al.
and the kinematic viscosity of the fluid. Experimental studies
carried out by Kulacki and Goldstein6 to characterize thermal
convection in a horizontal fluid layer with uniform volumetric heat source revealed that time-dependent mixing effects
begin to play the dominant role in the overall energy transport process for Rayleigh numbers 共Ra兲 greater than 104.
Therefore, the thermal convection currents in a Joule-heated
glass melt can be either steady or unsteady, depending on its
Rayleigh number.
Two different glasses with widely different infrared absorption properties—a semitransparent glass with a high effective thermal conductivity and a much more opaque glass
with low effective thermal conductivity—were used in this
study. The effective thermal conductivity combining the conductive and radiative heat transfer can be expressed as
FIG. 1. Computational domain and boundary conditions.
the numerical solution is discussed in Sec. III. Computational results and their discussion are given in Sec. IV, which
is followed by final conclusions in Sec. V.
II. PROBLEM FORMULATION
Ra =
g␤QL5
,
k␣␯
where the radiation coefficient r is expressed as
r=
A two-dimensional rectangular cavity with a horizontal
dimension of 0.2 m and a vertical dimension of 0.1 m was
chosen for investigating the influence of time-dependent
thermal forcing on chaotic mixing in an internally heated
glass melt. The problem to be considered here is defined in
Fig. 1, which illustrates the electrode configuration used for
Joule heating the electrically conducting molten glass in the
rectangular cavity.
Natural convection in an internally heated fluid with
constant physical properties in a rectangular cavity can be
characterized by the Rayleigh number, the Prandtl number,
and the aspect ratio. The Rayleigh number characterizing
the thermal convection in a uniformly heated fluid is expressed as
共1兲
where Q is the uniform volumetric heating power and L is
the characteristic length, generally corresponding to the
depth of the portion of the layer in unstable equilibrium.
Other parameters g, k, ␣, ␤, and ␯ are, respectively, the
gravitational acceleration, the thermal conductivity, the thermal diffusivity, the isobaric coefficient of thermal expansion,
共2兲
keff = k + rT3 ,
16n2␥
.
3KR
共3兲
In Eq. 共3兲, n is the refractive index of the glass, ␥ is the
Stefan–Boltzmann constant, and KR is the mean absorption
coefficient.9 The thermophysical properties of the semitransparent and the opaque glasses used for simulations are given
in Table I. The Rasch–Hinrichsen model and the Fulcheris
model are used, respectively, to account for the variation in
electrical conductivity and viscosity of molten glass with
temperature.9 Properties identical to those of the semitransparent glass were used for the opaque glass except the radiation coefficient r, which was set to zero assuming negligible
internal heat transfer by radiation in a nearly black glass
melt. Thus, the semitransparent glass has an effective thermal conductivity of 124.94 W/mK and the opaque glass a
thermal conductivity of 2.45 W/mK at 1525 K. Since the
physical properties such as viscosity, thermal conductivity,
and electrical conductivity vary in the glass melt, dimensional model equations were used to describe the glass melt
conditions.
The glass melt was assumed to be Newtonian.9 The glass
melt flow was computed by solving the unsteady Navier–
Stokes equation
TABLE I. Physical properties of the semitransparent and the opaque glasses used for simulations.
Property
Density 共␳ , kg/ m3兲
Coefficient of expansion 共␤, 1/K兲
Specific heat 共c P, J/kg K兲
Thermal conductivity 共k, W/m K兲
Radiation coefficient 共r , W / m K4兲
Viscosity 共␩, Pa s兲
Electrical conductivity 共␴, S/m兲
Semitransparent glass
Opaque glass
3328.7
10−4
1285
2 ⫻ 10−6T2 − 3.2⫻ 10−3T + 2.68
3.4538⫻ 10−8
9.92⫻ 10−2e1046.2/共T−1033.41兲
1.74⫻ 106e共−20 300/T兲
3328.7
10−4
1285
2 ⫻ 10−6T2 − 3.2⫻ 10−3T + 2.68
0
9.92⫻ 10−2e1046.2/共T−1033.41兲
1.74⫻ 106e共−20 300/T兲
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013101-3
Phys. Fluids 22, 013101 共2010兲
Chaotic mixing in a Joule-heated glass melt
FIG. 2. 共Color online兲 Initial distribution of passive tracer particles.
␳
⳵V
+ ␳V · ⵜV = − ⵜp + ⵜ · 关␩共T兲共ⵜV + ⵜVT兲兴
⳵t
+ ␳0g␤共T − T0兲,
共4兲
together with the incompressibility condition
ⵜ · V = 0,
共5兲
where V is the velocity vector; p is the pressure; g is the
acceleration due to gravity; and ␳, ␤, and ␩共T兲 are the density, coefficient of thermal expansion, and viscosity, respectively. The electromagnetic body force 共Lorentz兲 term was
not included in Eq. 共4兲 as the study was carried out without
any externally applied magnetic field and self-induced
Lorentz forces were neglected. A ratio of the electromagnetic
body force to the gravitational body force 关J0B0 / ␳0g␤
⫻共T − T0兲, where J0 and B0 are the characteristic electric current density and magnetic flux density, respectively兴 can be
used to compare the relative strength of the electromagnetic
body force with the gravitational body force for the present
problem. Since there is no externally applied magnetic field
and the self-induced magnetic field is negligibly small in
poorly electrically conducting, nonmagnetic, highly viscous
glass melts, J0B0 / ␳0g␤共T − T0兲 Ⰶ 1 and, therefore, the electromagnetic body force can be neglected.
The temperature T in the glass melt was obtained by
solving the energy conservation equation
␳c P
⳵T
J2
+ ␳c PV · ⵜT = ⵜ · 关keff共T兲 ⵜ T兴 +
,
⳵t
␴共T兲
FIG. 3. 共a兲 Isotherms and 共b兲 streamlines in the semitransparent glass melt
during steady thermal forcing with the middle electrode at 30 V, the top
electrode at 0 V, and both the left and right electrodes switched off.
glass melting operations, for which the electric field can be
decoupled from the flow field.9 The electric current density J
in the melt was computed by solving the Laplace equation
ⵜ · 关␴共T兲 ⵜ ␸兴 = 0,
共7兲
together with the Ohm’s law,
J = − ␴共T兲 ⵜ ␸ ,
共8兲
where ␸ is the electric scalar potential and ␴共T兲 is the electrical conductivity of the model fluid.
The internal heating was realized using a set of electrodes located at the horizontal boundaries. The electrode
共6兲
where the term J2 / ␴共T兲 is the volumetric heat source in
the glass melt on account of Joule heating. The viscous heating is very small compared with the Joule heat generation
in the glass and is neglected as the Brinkman number,
Br= ␩0V20 / k0⌬T, satisfies the condition BrⰆ 1.
Both the magnetic Reynolds number 共Rem = ␮0␴0V0L,
where ␮0 is the magnetic permeability and ␴0 is the electrical
conductivity兲 and the ratio of the length scale of the furnace
and wave length of the oscillating field 共Nf = ␻␮0␴0L2, where
␻ is the frequency of the electric field兲 are typically small in
FIG. 4. 共Color online兲 Time evolution of spread of passive tracers in the
semitransparent glass melt during steady thermal forcing with the middle
electrode at 30 V, the top electrode at 0 V, and both the left and right
electrodes switched off.
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FIG. 5. Electrode conditions for a typical electrode firing cycle with period
TC = 240 s during time-dependent thermal forcing. The top electrode is always kept at 0 V.
configuration used for the present study is shown in Fig. 1.
Numerical simulations were carried out for both steady and
time-dependent heating of the glass melt. The top electrode
covering the melt surface was invariably kept at zero potential in all the simulations. Out of the three strip electrodes
located at the bottom boundary, the middle electrode was
switched on and maintained at a constant electric potential,
while both the left and right electrodes were switched off
during the steady thermal forcing. All the three electrodes
located at the bottom boundary were periodically energized
during the time-dependent thermal forcing. The boundary
conditions used for the study are explained in Fig. 1. All
gradients in the y-direction were neglected to carry out the
two-dimensional simulation in the x-z plane.
FIG. 6. Transient behavior of various quantities at the center 共x = 0.1 m,
z = 0.05 m兲 of the semitransparent melt during time-dependent thermal forcing with firing cycle period TC = 240 s. 共a兲 Joule heat generation, 共b兲 temperature, and 共c兲 x-component of the velocity vector.
angular boxes, each having an area of 2.5⫻ 1.25 mm2, was
used such that 98% of the boxes would contain at least one
out of the 25 600 randomly distributed particles. A degree of
mixing based on the two species was used to quantify the
stirring efficiency in the Joule-heated glass melt.
The degree of mixing based on two species can be defined as4
Dm =
S
Smax
where S is the mixing entropy of the particles given by the
expression
K
III. NUMERICAL SOLUTION
冉
2
冊
S = − 兺 wi 兺 关ni,j log共ni,j兲兴 ,
i=1
The model equations, Eqs. 共4兲–共8兲, were solved using
the commercial general-purpose ccomputational fluid dynamics software FLUENT. In order to study the chaotic advection in the glass melt, two strips of passive tracer particles
were tracked employing the discrete phase model of
10
FLUENT.
The initial locations of the passive tracers are
shown in Fig. 2. A left strip of 25 600 particles distributed
horizontally between points B1 共x = 0.005 m, z = 0.05 m兲
and B2 共x = 0.095 m, z = 0.05 m兲 and a right strip of 25 600
particles distributed vertically between points R1
共x = 0.15 m, z = 0.005 m兲 and R2 共x = 0.15 m, z = 0.095 m兲
were tracked using the Runge–Kutta scheme. Since all the
simulations were carried out using isothermal, static flow
field 共all velocity components equal to zero兲 as the initial
condition, the numerical simulations were continued for sufficient period to obtain fully developed thermal and flow
fields with respect to time. The material lines were introduced in the flow domain subsequently.
The box counting method was used to investigate the
extent of advection due to the natural convection in the glass
melt. In the box counting method, the box size is related to
the total number of particles tracked.11 A total of 6400 rect-
共9兲
,
j=1
共10兲
and Smax is the maximum obtainable entropy for the twospecies system given by the expression
Smax = − K log共 21 兲 .
共11兲
In Eqs. 共10兲 and 共11兲, K is the total number of rectangular
boxes, ni,j is the fraction of species j in the ith box, and wi is
a weighting function with a value of zero in boxes that contain only a single species, and unity in boxes that contain
both the species. A value of Dm = 1 suggests a perfectly uniform mixing while Dm = 0 indicates a complete segregation
of the two species.
IV. RESULTS
A. Chaotic mixing in a semitransparent glass melt
The results obtained for the semitransparent glass with a
high effective thermal conductivity are presented in this section. Figure 3 shows isotherms and streamlines computed for
steady thermal forcing, which was accomplished by applying
a steady voltage of 30 V to the middle electrode and keeping
the top electrode grounded. Both the left and right electrodes
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013101-5
Chaotic mixing in a Joule-heated glass melt
Phys. Fluids 22, 013101 共2010兲
FIG. 7. Snapshots of 共a兲 isotherms and 共b兲 streamlines in the semitransparent glass melt during time-dependent thermal forcing with firing cycle period TC
= 240 s.
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Phys. Fluids 22, 013101 共2010兲
FIG. 8. 共Color online兲 Time evolution of spread of passive tracers in the
semitransparent glass melt during time-dependent thermal forcing with firing cycle period TC = 240 s.
were not energized in this case. The predicted flow and thermal fields were steady and the computed average power density in the glass melt was 1.7⫻ 105 W / m3. Relatively large
electric potential gradients generated near the middle electrode gave rise to a higher local current density. The nonuniform volumetric heat generation set up thermal convection
currents with two counter-rotating flow cells producing a
central upward flow. Each of the two recirculation cells was
comprised of an elliptic point surrounded by closed streamlines. The predicted isotherms showed that conduction had a
significant role in the overall heat transport process for the
semitransparent glass. The internal heat transfer by radiation
in the semitransparent glass resulted in a large effective thermal conductivity, and thereby reduced the driving force for
the thermal convection substantially.
Deformations of the left and right passive material strips,
with initial conditions as shown in Fig. 2, induced by the
thermal convection currents obtained for the steady thermal
forcing case are shown in Fig. 4. The time snapshots clearly
indicate that the material lines are stretched along the streamlines such that particles of the left and right strips are confined to the left and right halves, respectively. The weighting
function wi is zero for all boxes in this case, and as a
result the degree of mixing Dm is zero throughout the time
evolution.
The mixing behavior of the steady thermal convection in
the Joule-heated glass melt can be enhanced substantially by
means of chaotic advection induced by time-dependent thermal forcing. Mixing is enhanced when the time-dependent
streamlines intersect transversely at disjoint regions. Aref’s
blinking vortex system, in which streamlines caused by two
blinking vortices intersect each other, is a well known example of this.12 A typical electrode firing cycle used for the
simulation of time-dependent thermal forcing is shown in
Fig. 5. In this cycle, each electrode was fired for TC / 4 s
using a sinusoidal voltage ␸ = 冑2 ⫻ 30 sin共4␲t / Tc兲 periodically in the order left-middle-right-middle with a period of
TC = 240 s. The Joule heat generation 共Q兲, temperature 共T兲,
and x-component 共u兲 of the velocity vector computed at the
center of the cavity 共x = 0.1 m, z = 0.05 m兲 for one firing
cycle with TC = 240 s are shown in Fig. 6. The predicted
flow and thermal transients clearly show a periodicity iden-
FIG. 9. 共Color online兲 Comparison of spread of passive tracers in the semitransparent glass melt at t = 1920 s for different electrode firing cycle periods, TC.
tical to that of the electrode firing cycle. The thermal transient in the melt also exhibits a phase shift in comparison to
the Joule heating cycle indicating the system inertia. The
oscillating horizontal velocity at the center of the cavity
around a mean value of u = 0 m / s suggests the presence of
growing and shrinking flow cells in the cavity. A negative
value for u at the center means a relatively larger anticlockwise vortex, while a positive value means a relatively larger
clockwise vortex. As seen in Fig. 6, the zero crossing of the
horizontal velocity is enforced by firing the left and right
electrodes. Local heating by these electrodes suppresses the
nearby vortex and allows the other vortex to grow. This feature is further illustrated in Fig. 7, which shows the snapshots of isotherms and streamlines for the electrode firing
cycle with TC = 240 s. The t = 0 s frames show the isotherms
and streamlines corresponding to the end of the previous
firing cycle. At this stage, the flow pattern comprises of a
relatively larger anticlockwise vortex. Local heating by the
left electrode during the first quarter of the current firing
cycle initiates a braking effect on the anticlockwise vortex,
which, therefore, starts shrinking 共see t = 40 s frames兲. This
continues until the middle electrode heating during the second quarter nullifies the braking effect 共see t = 80 s frames兲.
At the end of the second quarter, t = 120 s, mirror images of
the t = 0 s frames exist 共similar to the relation between
frames t = 40 s and t = 160 s兲. Similar trend is observed in
the next half of the firing cycle 共see t = 160 s and t = 200 s兲.
The computed flow patterns indicate the occurrence of timedependent streamlines forming transversely intersecting annuli and, thereby, producing well-mixed regions.
The time evolution of the passive tracers obtained for the
time-dependent thermal forcing with a period of TC = 240 s
is shown in Fig. 8. Unlike in the steady-thermal forcing case,
trajectories of particles of both the left and right material
lines exhibit chaotic behavior when the electrodes are periodically fired. The snapshot at t = 240 s shows that the left
material line is stretched to reach the right half, while the
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Chaotic mixing in a Joule-heated glass melt
Phys. Fluids 22, 013101 共2010兲
FIG. 10. Comparison of streamlines in the semitransparent glass melt in the beginning 共t / TC = 0兲 of the firing cycle for different electrode firing cycle
periods, TC.
right material line is stretched to reach the left half of the
cavity. As the time proceeds, repeated stretching and folding
result in the formation of a horseshoe structure leading to a
well-mixed glass melt except near the boundaries. Generally
a system can be classified as chaotic if the flow produces
horseshoe maps.13
In order to demonstrate the impact of the TC on chaotic
mixing, snapshots of passive tracers at t = 1920 s for different values of firing cycle period—TC = 0, 30, 60, 120, 240,
and 360 s—are compared in Fig. 9. The frame TC = 0 s corresponds to the steady thermal forcing case with middle electrode firing. 共For time-dependent thermal forcing, TC = 0 s
implies simultaneous firing of all the electrodes. Since the
Joule heating on account of the side electrodes is substantially low compared with that of the middle electrode
关see Fig. 6共a兲兴, flow patterns for steady and time-dependent
thermal forcing cases look alike for TC = 0 s.兲 The frame
TC = 30 s shows that the material lines are stretched and
folded repeatedly at the outer region of the vortex creating
chaotic Lagrangian motion around an inner regular area.
There is no intermixing of the particles of the left and right
material lines for TC = 30 s. Increasing the time period to
TC = 60 s not only enlarges the chaotic region but also results
in intermixing of the particles of the left and right material
lines. Snapshots at TC = 120 s and TC = 240 s reveal progressive reduction in the area of the regular islands with increasing TC. However, in the frame TC = 360 s, striations are
clearly visible suggesting a reduction in the degree of mixing
compared with that of the TC = 240 s case.
Streamlines in the beginning 共t / TC = 0兲 of the firing cycle
for different cycle periods—TC = 30, 60, 120, and 360 s—are
presented in Fig. 10 so as to explain the difference in mixing
behavior observed in Fig. 9. The TC = 30 s frame illustrates
the insignificant transverse intersection of the clockwise and
anticlockwise vortices, and thereby explains the absence of
intermixing of the particles of the left and right material
lines. The chaotic Lagrangian motion observed in the outer
region of the vortex is due to the transverse intersection of
the streamlines of the same vortex with its own time maps.
For example, streamlines at 共t / TC = 0.5兲 are mirror images of
those at 共t / TC = 0兲 and they intersect transversely at the outer
region of the vortex. The area of this transverse intersection
increases with TC. The stirring mechanism is similar to that
of the rotational stretching and folding produced by an oscillating point vortex.14 However, the present case involves two
oscillating vortices with varying size and circulation. The
degree of mixing improves with the increase in the area of
transverse intersection of streamlines.15 Figure 10 also indicates that the area of the transverse intersection of the clockwise and anticlockwise vortices increases with TC and,
thereby, justifies the enhanced intermixing of the particles of
the left and right material lines.
The impact of the side electrode firing can be well understood from Fig. 11, which plots the deviation from the
steady vortex turnover time. In the case of steady thermal
forcing using the middle electrode alone 关see Fig. 3共b兲兴, the
values of the vortex turnover time calculated along the
streamlines at 共x = 0.01 m, z = 0.05 m兲, which belongs to the
left 共anticlockwise兲 vortex, and 共x = 0.19 m, z = 0.05 m兲,
which belongs to the right 共clockwise兲 vortex, are identical
共=269.5 s兲. In Fig. 11, solid lines show the deviation from
the steady vortex turnover time for the left vortex and the
dashed lines for the right vortex. A negative deviation implies a reduction in the vortex size, whereas a positive deviation means an enlargement. Figure 11共a兲 shows the same
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013101-8
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Phys. Fluids 22, 013101 共2010兲
FIG. 11. Comparison of deviation from steady vortex turnover time for different electrode firing cycle periods, TC 共semitransparent glass melt兲.
trend in the reduction and enlargement in the vortex size for
different TC values 共except TC = 360 s兲. The amplitude of the
deviation from the steady vortex turnover time increases
with TC. This implies that the area of transverse intersection
of the left and right vortices and, thereby, chaotic mixing
increases with TC. Figure 11共b兲 compares the deviation from
the steady vortex turnover time for TC = 360 s with that for
TC = 240 s. The behavioral difference between the two cases
can be easily understood by means of Fig. 12, which shows
the streamlines for different values of t / TC 共=0, 1/6, 2/6, and
3/6兲 for TC = 360 s. The prolonged firing of the side electrodes, in the case of TC = 360 s, leads to strong braking
effect on the nearby vortex and, thereby, reduces its size
substantially such that only one primary vortex remains at
some point of time during the subsequent middle electrode
firing 共see t / TC = 1 / 6 and 2/6 frames兲. The overall transverse
intersection of the time-dependent streamlines for the entire
cycle seems to be inferior for TC = 360 s when compared
with that of TC = 240 s. The Poincaré maps reported for
blinking flow with two superimposed streamline patterns
FIG. 12. Snapshots of streamlines in the semitransparent glass melt during time-dependent thermal forcing with firing cycle period TC = 360 s.
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TC = 240 s, after which it begins to decline due to reduction
in overall transverse intersection of the time-dependent
streamlines, as explained earlier.
B. Chaotic mixing in an opaque glass melt
FIG. 13. Comparison of local stretch ratio in the semitransparent glass melt
during time-dependent thermal forcing for different electrode firing cycle
periods, TC.
with a separatrix at different locations in a rectangular cavity
corroborate the features of chaotic mixing observed for different values of TC as in Fig. 9.16
The results of the stretch ratio calculations carried out
for different values of TC are presented in Fig. 13. A string
of 1000 particles uniformly distributed between the points
共x = 0.024 m, z = 0.05 m兲 and 共x = 0.026 m, z = 0.05 m兲 was
tracked for 3600 s to obtain the stretch ratios. The steady
thermal forcing case, as expected, exhibits a linear growth.
The local stretch ratio grows faster as TC increases and
reaches an exponential growth for TC = 120 s indicating a
globally chaotic flow.
A comparison of the evolution of the degrees of mixing
for different firing cycle periods is presented in Fig. 14共a兲.
Figure 14共b兲 shows the effect of TC on the degree of mixing
and the local stretch ratio. The plots show that the degree of
mixing increases with TC up to a peak value Dm = 0.85 at
In order to study the extent of chaotic advection in the
opaque glass, a steady thermal forcing case was considered
identical to that of the semitransparent glass. For the opaque
glass, the steady thermal forcing simulation was carried out
by applying a steady voltage of 25 V to the middle electrode
and keeping the top electrode grounded. The computed average power density in the glass melt was 1.7⫻ 105 W / m3.
Unlike in the semitransparent glass, the predicted flow and
thermal fields exhibited time-dependent oscillations. This
was expected in the case of the opaque glass as its poor
effective thermal conductivity gave rise to strong driving
force for thermal convection. Time snapshots of the isotherms in the opaque glass and the corresponding streamlines
are shown in Fig. 15. The predicted isotherms show the
downward release of cold thermal spike due to the Rayleigh–
Taylor 共RT兲 instability from the edge of the thermal boundary layer at the top surface of the glass melt. A similar behavior was also observed in the experimental study of
Kulacki and Goldstein6 to characterize thermal convection in
a horizontal fluid layer with uniform volumetric heat source.
RT turbulent mixing occurs in diverse phenomena ranging
from temperature inversions in the atmosphere to supernova
explosions. However, mixing in highly viscous glass melts is
believed to be on account of the chaotic motion caused by
the time-dependent, two-dimensional flow field. Model predictions show that the degree of mixing 共Dm兲 in the opaque
glass during steady thermal forcing asymptotically approaches a value of 0.87.
A supplementary simulation was carried out for the
opaque glass to study the effect of time-dependent thermal
forcing using periodically fired electrode configuration
FIG. 14. 共a兲 Time evolution of degree of mixing and 共b兲 TC-dependence of local stretch ratio and degree of mixing in the semitransparent glass melt during
time-dependent thermal forcing.
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013101-10
Gopalakrishnan et al.
Phys. Fluids 22, 013101 共2010兲
FIG. 17. Power spectra of u—velocities plotted in Fig. 16: 共a兲 for steady
thermal forcing and 共b兲 for time-dependent thermal forcing using a firing
cycle with TC = 240 s.
FIG. 15. 共a兲 Isotherms and 共b兲 streamlines in the opaque glass melt during
steady thermal forcing with the middle electrode at 25 V, the top electrode at
0 V, and both the left and right electrodes switched off.
identical to the one used for the semitransparent glass. During the firing cycle, similar to that shown in Fig. 5, each
electrode was periodically fired for 60 s using a sinusoidal
voltage ␸ = 冑2 ⫻ 25 sin共4␲t / Tc兲 with the firing cycle period
TC = 240 s. The transient behaviors of x-component 共u兲 of
the velocity vector computed at the center 共x = 0.1 m,
z = 0.05 m兲 of the cavity during the steady and the timedependent thermal forcing conditions are compared in Fig.
16 and their power spectra are compared in Fig. 17. The
power spectrum of the steady thermal forcing case shows the
existence of a single dominant periodic component, whereas
that of the time-dependent thermal forcing case indicates the
FIG. 16. Transient behavior of the x-component 共u兲 of the velocity vector
computed at the center 共x = 0.1 m, z = 0.05 m兲 of the opaque glass melt
during 共a兲 steady thermal forcing and 共b兲 time-dependent thermal forcing
using a firing cycle with TC = 240 s.
presence of multiple dominant frequencies showing more
complex time dependence. Snapshots of the streamlines and
time evolution of spread of passive tracers corresponding to
the two cases are compared in Figs. 18 and 19, respectively.
Like in the semitransparent glass melt, the side electrode
firing resulted in time-dependent changes in the size of the
primary vortices. However, unlike in the semitransparent
glass melt, thermal plumes released from the top and bottom
thermal boundary layers significantly modified the shape of
these vortices. Figure 18 shows that the transverse intersection of the time-dependent streamlines is better in the case of
time-dependent thermal forcing. Figure 19 indicates that
stretching and folding are substantially enhanced by the periodic firing of the electrodes in comparison to the steady
thermal forcing case. This observation is in concurrence with
Fig. 20 which compares the degrees of mixing for the two
cases. In the steady thermal forcing case, Dm = 0.85 was
achieved in 2640 s, whereas the same was accomplished in
1400 s in the case of time-dependent thermal forcing employing periodically fired electrodes with TC = 240 s.
V. SUMMARY AND CONCLUSIONS
Chaotic mixing in the Joule-heated glass melt in a twodimensional rectangular cavity with a horizontal dimension
of 0.2 m and a vertical dimension of 0.1 m was numerically
investigated using relative dispersion of Lagrangian tracers.
Two different glasses—a semitransparent glass and an
opaque glass—were used to compare their mixing behaviors.
The concept of mixing entropy was used to quantify the
degree of mixing in the glass melt. The present study shows
that steady thermal convection occurs in the semitransparent
glass during the steady thermal forcing with an average
power density of 1.7⫻ 105 W / m3. The model predictions
indicate that the chaotic mixing induced by time-dependent
thermal forcing using periodically fired electrodes is strongly
dependent on the period of electrode firing cycle TC. For
small TC, there is no significant transverse intersection of the
clockwise and anticlockwise vortices and the material lines
are stretched and folded repeatedly only at the outer region
of the vortex, creating chaotic Lagrangian motion around an
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013101-11
Chaotic mixing in a Joule-heated glass melt
Phys. Fluids 22, 013101 共2010兲
FIG. 18. Comparison of streamlines in the opaque glass melt: 共a兲 steady thermal forcing and 共b兲 time-dependent thermal forcing using a firing cycle with
TC = 240 s.
inner regular area. The chaotic Lagrangian motion observed
in the outer region of the vortex is due to the transverse
intersection of the streamlines of the same vortex with its
own time maps. Increase in TC results in transverse intersection of the clockwise and anticlockwise vortices and progressive reduction in the area of the regular islands. However,
striations are clearly visible after an optimum TC. The stirring mechanism involves the rotational stretching and folding produced by two oscillating vortices with varying size
and circulation. Numerical results for the semitransparent
glass show that the degree of mixing increases with TC up to
a peak value of Dm = 0.85 at TC = 240 s, after which it begins
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013101-12
Phys. Fluids 22, 013101 共2010兲
Gopalakrishnan et al.
FIG. 20. Comparison of degree of mixing in the opaque glass for steady
thermal forcing using middle electrode firing and time-dependent thermal
forcing using three electrode configuration with a firing cycle with
TC = 240 s.
ACKNOWLEDGMENTS
FIG. 19. 共Color online兲 Comparison of spread of passive tracers in the
opaque glass: 共a兲 steady thermal forcing and 共b兲 time-dependent thermal
forcing using a firing cycle with TC = 240 s.
This work was supported by the Deutsche Forschungsgemeinschaft under Contract No. TH497/26-1. The authors
are grateful to Henning Schwanbeck for the valuable support
from the computer center, TU Ilmenau.
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2
to decline due to reduction in overall transverse intersection
of the time-dependent streamlines.
Numerical simulation using the opaque glass shows that
time-dependent flow occurs for steady thermal forcing with
an average power density of 1.7⫻ 105 W / m3. The naturally
occurring time-dependent flow gives rise to chaotic advection in the opaque glass. This study shows that a degree of
mixing Dm = 0.85 is accomplished in 2640 s due to the naturally occurring chaotic advection. However, time-dependent
thermal forcing employing periodically fired multiple electrode configuration improves mixing in the opaque glass
melt substantially. For example, the same degree of mixing
Dm = 0.85 can be accomplished in 1400 s using periodically
energized electrodes with TC = 240 s.
Thus, the present study clearly establishes that mixing
behavior of a Joule-heated glass melt flow can be enhanced
substantially by introducing external time dependency
through time-dependent thermal forcing employing periodically energized multiple electrode configuration. Although
this study was carried out for a given electrode configuration
for inducing external time dependency in the flow, the numerical predictions clearly indicate the possibility of optimizing the electrode configuration and the firing pattern to
obtain the best possible thermal and chemical homogenization of the glass melt.
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